Superalgebras have emerged in a wide variety of areas ranging from high energy and condensed matter physics such as topological field theory, logarithmic conformal field theories (CFTs), the integer quantum Hall transition, sigma models on supergroup manifolds and superstring theory (the only availabe candidate for unifying all four forces in nature), just to mention a few. In particular, the replica or supersymmetric treatment of disordered systems has revealed that the random critical points in two dimensions are described by current superalgebras with zero superdimension and their corresponding non-unitary CFTs.
This project aims to investigate such current superalgebras and
CFTs as well as their applications in disordered systems and the integer
quantum Hall transition.
The study of correlation functions is the major open problem in the theory of exactly soluble models in condensed matter physics and integrable quantum field theories. There is much interest in this field internationally, leading to high-profile research activity at present. A key part of this project is to evaluate correlation functions and form factors of physical significance. The recent development of sophisticated algebraic and analytic methods makes the computation of such correlators feasible.
A great success in the understanding of physical phenomena such as phase transitions in quantum systems has come through the study of exactly soluble models. As is well known, most problems of physical interest are of a non-perturbative and non-linear nature and therefore are very difficult to solve. Under proper approximations, these complex physical problems reduce to less complex but physically non-trivial models that can be solved exactly. Such exactly soluble models inherit many physical features of, and therefore provide important insight into, the original systems. One of the consequences of such approximations is that form factors and correlation functions can be computed in closed form, so providing essential information for the full description of the original systems.
This project is to capitalize on our recent success in the evaluation
of correlators of some soluble models by undertaking a through and systematic
investigation into the algebraic formulation of correlation functions and form
factors. This is to be achieved by further developing the vertex operator
method as well as the factorizing F-matrix method. The latter approach will
enable correlation functions and form factors to be expressed in determinant forms.
Quasi exact solvability is closely related to exact solvability. If all the eigenvalues of a quantum mechanical system are known together with the corresponding eigenfunctions the system is exactly solvable. In contrast a system is quasi exactly solvable if only a finite number (usually the lowest lying ones) of exact eigenvalues and eigenfunctions are known. Many quasi exactly solvable systems of physical significance have been identified either by algebraization or by deforming an exactly solvable systems with an addition of a higher order interaction term together with a compensation term. Although solutions to quasi exactly solvable systems of one degree of freedom are reasonably simple to obtain by i.e. the Bethe ansatz method among other methods it has been a challenge to find exact solutions (exact eigenvalues and eigenfunctions within the invariant subspace) to quasi exactly solvable systems of many degrees of freedom.
The overall objectives of this project are to investigate mathematical structures of the multi-particle quasi exactly solvable systems and develop techniques for solving these systems exactly. This is a project in collaboration with Professor Ryu Sasaki at Yukawa Institute for Theoretical Physics of Kyoto University. We are looking for students to join us to work on the project.